An important question is to check whether such an optimal map exists. If \(\)  is discrete but not \(\) , then no Monge map exists, since \(T_\sharp \)  is necessarily discrete, so that the mass conservation constraint set \(T_\sharp =\)  is empty. Brenier's theorem [1]} ensures the existence and uniqueness of a Monge map in \(^d\) , for the cost \(x,y)={x-y}^2\) , if \(\)  has a density with respect to Lebesgue's measure. Furthermore the Monge map is the unique gradient of a convex function satisfying the mass conservation constraint, i.e. \(T=\nabla u\)  with \(u\)  a convex function.
